Fermion masses and mixings in the 3-3-1 model with right ...634 Page 2 of 19 Eur. Phys.. CJ (2016) 76 :634 Table 1 Range for experimental values of neutrino mass squared splittings

Eur. Phys. J. C (2016) 76:634DOI 10.1140/epjc/s10052-016-4480-3

Regular Article - Theoretical Physics

Fermion masses and mixings in the 3-3-1 model with right-handedneutrinos based on the S3 flavor symmetry

A. E. Cárcamo Hernández1,a, R. Martinez2,b, F. Ochoa2,c

1 Universidad Técnica Federico Santa María, Casilla 110-V, Valparaiso, Chile2 Departamento de Física, Universidad Nacional de Colombia, Ciudad Universitaria, Bogotá, D.C., Colombia

Received: 25 February 2016 / Accepted: 6 November 2016 / Published online: 21 November 2016© The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract We propose a 3-3-1 model where the SU (3)C ⊗SU (3)L ⊗ U (1)X symmetry is extended by S3 ⊗ Z3 ⊗Z ′

3 ⊗ Z8 ⊗ Z16 and the scalar spectrum is enlarged byextra SU (3)L singlet scalar fields. The model successfullydescribes the observed SM fermion mass and mixing pat-tern. In this framework, the light active neutrino masses arisevia an inverse seesaw mechanism and the observed chargedfermion mass and quark mixing hierarchy is a consequenceof the Z3 ⊗ Z ′

3 ⊗ Z8 ⊗ Z16 symmetry breaking at very highenergy. The obtained physical observables for both quark andlepton sectors are compatible with their experimental values.The model predicts the effective Majorana neutrino massparameter of neutrinoless double beta decay to be mββ = 4and 48 meV for the normal and the inverted neutrino spectra,respectively. Furthermore, we found a leptonic Dirac CP-violating phase close to π

2 and a Jarlskog invariant close toabout 3 × 10−2 for both normal and inverted neutrino masshierarchy.

1 Introduction

After the discovery of the 126 GeV Higgs boson by ATLASand CMS collaborations at CERN Large Hadron Collider(LHC) [1,2], the vacancy of the Higgs boson needed for thecompletion of the Standard Model (SM) at the Fermi scalehas been filled and the weak gauge bosons mass generationmechanism has also been confirmed. Despite LHC experi-ments indicating that the decay modes of the new scalar stateare SM like, there is still room for new extra scalar states,whose search are an essential task of the LHC experiments.Furthermore, despite the great consistency of the SM pre-dictions with the experimental data, there are several aspects

a e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]

that the SM does not explain, some of them are the observedhierarchy among charged fermion masses and quark mix-ing angles, the tiny neutrino masses and the smallness of thequark mixing angles, which contrast with the sizable leptonicmixing ones. The global fits of the available data from theDaya Bay [3], T2K [4], MINOS [5], Double CHOOZ [6]and RENO [7] neutrino oscillation experiments, constrainthe neutrino mass squared splittings and mixing parame-ters [8]. It is a well-established experimental fact that theobserved hierarchy of charged fermion masses goes over arange of five orders of magnitude in the quark sector and thatthere are six orders of magnitude between the neutrino massscale and the electron mass. Accommodating the chargedfermion masses in the SM requires an unnatural tunningamong its different Yukawa couplings. Furthermore, experi-ments with solar, atmospheric and reactor neutrinos [3–7,9]have brought about evidence of neutrino oscillations causedby nonzero masses. All these unexplained issues stronglyindicate that new physics has to be invoked to address thefermion puzzle of the SM (Table 1).

The aforementioned flavor puzzle, not understood in thecontext of the SM, motivates extensions of the StandardModel that explain the fermion mass and mixing patterns.From the phenomenological point of view, it is possible todescribe some features of the mass hierarchy by assumingYukawa matrices with texture zeros [10–38]. A very promis-ing approach is the use of discrete flavor groups, which havebeen considered in several models to explain the fermionmasses and mixing (see Refs. [39–42] for recent reviews onflavor symmetries). Models with spontaneously broken flavorsymmetries may also produce hierarchical mass structures.Recently, discrete groups such as A4 [43–62], S3 [63–83],S4 [84–94], D4 [95–104], Q6 [105–108], T7 [109–118], T13

[119–122], T ′ [123–129], �(27) [130–144] and A5 [145–155] have been considered to explain the observed pattern offermion masses and mixings. In particular the S3 flavor sym-metry is a very good candidate for explaining the prevailing

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Table 1 Range for experimentalvalues of neutrino mass squaredsplittings and leptonic mixingparameters, taken from Ref. [8],for the case of normal hierarchy

Parameter �m221(10−5eV2) �m2

31(10−3eV2)(sin2 θ12

)exp

(sin2 θ23

)exp

(sin2 θ13

)exp

Best fit 7.60 2.48 0.323 0.567 0.0234

1σ range 7.42–7.79 2.41–2.53 0.307–0.339 0.439–0.599 0.0214–0.0254

2σ range 7.26–7.99 2.35–2.59 0.292–0.357 0.413–0.623 0.0195–0.0274

3σ range 7.11–8.11 2.30–2.65 0.278–0.375 0.392–0.643 0.0183–0.0297

Table 2 Range for experimentalvalues of neutrino mass squaredsplittings and leptonic mixingparameters, taken from Ref. [8],for the case of inverted hierarchy

Parameter �m221(10−5eV2) �m2

13(10−3eV2)(sin2 θ12

)exp

(sin2 θ23

)exp

(sin2 θ13

)exp

Best fit 7.60 2.38 0.323 0.573 0.0240

1σ range 7.42–7.79 2.32–2.43 0.307–0.339 0.530–0.598 0.0221–0.0259

2σ range 7.26–7.99 2.26–2.48 0.292–0.357 0.432–0.621 0.0202–0.0278

3σ range 7.11–8.11 2.20–2.54 0.278–0.375 0.403–0.640 0.0183–0.0297

pattern of fermion masses and mixing. The S3 discrete sym-metry is the smallest non-Abelian discrete symmetry grouphaving three irreducible representations (irreps), explicitlytwo singlets and one doublet irreps. The S3 discrete symme-try was used as a flavor symmetry for the first time in Ref.[156]. The different models based on discrete flavor sym-metries have as a common issue the breaking of the flavorsymmetry so that the observed data be naturally produced.The breaking of the flavor symmetry takes place when thescalar fields acquire vacuum expectation values (Table 2).

Besides that, another of the greatest mysteries in particlephysics is the existence of three fermion families at low ener-gies. The origin of the family structure of the fermions canbe addressed in family dependent models where a symmetrydistinguish fermions of different families. One explanationto this issue can be provided by the models based on thegauge symmetry SU (3)c ⊗ SU (3)L ⊗U (1)X , also called 3-3-1 models, which introduce a family non-universal U (1)Xsymmetry [26,61,62,78,79,115,117,157–202]. These mod-els have a number of phenomenological advantages. Firstof all, the three family structure in the fermion sector can beunderstood in the 3-3-1 models from the cancellation of chiralanomalies and asymptotic freedom in QCD. Second, the factthat the third family is treated under a different representationcan explain the large mass difference between the heaviestquark family and the two lighter ones. Third, these mod-els contain a natural Peccei–Quinn symmetry, necessary tosolve the strong-CP problem [203–206]. Finally, 3-3-1 mod-els including heavy sterile neutrinos have cold dark mattercandidates as weakly interacting massive particles (WIMPs)[179,207–209]. Besides that, the 3-3-1 models can explainthe 2 TeV diboson excess found by ATLAS [210]. Whenthe electric charge in the 3-3-1 models is defined in the lin-ear combination of the SU (3)L generators T3 and T8, it is afree parameter, independent of the anomalies (β). The choiceof this parameter defines the charge of the exotic particles.Choosing β = − 1√

3, the third component of the weak lepton

triplet is a neutral field νCR , which allows one to build theDirac matrix with the usual field νL of the weak doublet. Ifone introduces a sterile neutrino NR in the model, then it ispossible to generate light neutrino masses via inverse see-saw mechanism. The 3-3-1 models with β = − 1√

3have the

advantage of providing an alternative framework to generateneutrino masses, where the neutrino spectrum includes thelight active sub-eV scale neutrinos as well as sterile neutri-nos which could be dark matter candidates if they are lightenough or candidates for detection at the LHC, if their massesare at the TeV scale. This interesting feature makes the 3-3-1 models very interesting since if the TeV scale sterileneutrinos are found at the LHC, these models can be verystrong candidates for unraveling the mechanism responsiblefor electroweak symmetry breaking.

In the 3-3-1 models, one heavy triplet field with a VacuumExpectation Value (VEV) at high energy scale νχ , breaksthe symmetry SU (3)L ⊗ U (1)X into the SM electroweakgroup SU (2)L⊗U (1)Y , while the another two lighter tripletswith VEVs at the electroweak scale υρ and υη, trigger theelectroweak symmetry breaking [26]. Besides that, the 3-3-1 model could possibly explain the excess of events inthe h → γ γ decay, recently observed at the LHC, sincethe heavy exotic quarks, the charged Higges, and the heavycharged gauge bosons contribute to this process. On the otherhand, the 3-3-1 model reproduces a specialized Two HiggsDoublet Model type III (2HDM-III) in the low energy limit,where both electroweak triplets ρ and η are decomposed intotwo hypercharge-one SU (2)L doublets plus charged and neu-tral singlets. Thus, like the 2HDM-III, the 3-3-1 model canpredict huge flavor changing neutral currents (FCNC) andCP-violating effects, which are severely suppressed by exper-imental data at electroweak scales. In the 2HDM-III, for eachquark type, up or down, there are two Yukawa couplings.One of the Yukawa couplings is for generating the quarkmasses, and the other one produces the flavor changing cou-plings at tree level. One way to remove both the huge FCNC

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and CP-violating effects is by imposing discrete symmetries,obtaining two types of 3-3-1 models (type I and II models),which exhibit the same Yukawa interactions as the 2HDMtype I and II at low energy where each fermion is coupledat most to one Higgs doublet. In the 3-3-1 model type I, oneHiggs electroweak triplet (for example, ρ) provide masses tothe phenomenological up- and down-type quarks, simulta-neously. In the type II, one Higgs triplet (η) gives masses tothe up-type quarks and the other triplet (ρ) to the down-typequarks [26].

It is noteworthy the S3 flavor symmetry was implementedfor the first time in the 3-3-1 model of Ref. [69]. Thatmodel introduces a new U (1)L lepton global symmetry,responsible for lepton number and lepton parity. That lep-ton parity symmetry suppresses the mixing between ordinaryquarks and exotic quarks. Furthermore, the U (1)L new lep-ton global symmetry enforces to have different scalar fieldsin the Yukawa interactions for charged lepton, neutrino andquark sectors. The scalar sector of that model includes sixSU (3)L scalar triplets and four SU (3)L scalar antisextets.The SU(3)C ⊗ SU(3)L ⊗U (1)X ⊗U (1)L ⊗ S3 assignmentsof the fermion sector of the aforementioned model, requirethat these 6 SU (3)L scalar triplets be distributed as follows:3 for the quark sector, 2 for the charged lepton sector and1 for the neutrino sector. Furthermore the 4 SU (3)L scalarantisextets are needed to implement a type II seesaw mech-anism. In that model, light active neutrino masses are gener-ated from type-I and type-II seesaw mechanisms, mediatedby three heavy right-handed Majorana neutrinos and fourSU (3)L scalar antisextets, respectively. Since the Yukawaterms of that model are renormalizable, to explain the SMcharged fermion mass pattern, one needs to impose a stronghierarchy among the charged fermion Yukawa couplings ofthe model. Furthermore, the work described in Ref. [69] ismainly focused on the lepton sector, while in the quark sec-tor, the obtained quark mass matrices are diagonal and thequark mixing matrix is trivial.

Recently two of us proposed a SU (3)C × SU (3)L ×U (1)X ⊗ S3 ⊗ Z2 ⊗ Z4 ⊗ Z12 model [78], with a scalarsector composed of three SU (3)L scalar triplets and sevenSU (3)L scalar singlets, that successfully accounts for quarkmasses and mixings. In that model, all observables in thequark sector are in excellent agreement with the experimen-tal data, excepting

∣∣Vtd∣∣, which turns out to be larger by a

factor ∼1.3 than its corresponding experimental value, andnaively deviated 8 sigma away from it. That model has thefollowing drawbacks:

∣∣Vtd∣∣ is deviated 8 sigma away from its

experimental value, a S3 soft breaking term has to be intro-duced by hand in the low energy scalar potential in order tofullfill its minimization equations, the top quark mass arisesfrom a five dimensional Yukawa term and lepton masses andmixings are not addressed.

It is interesting to find an alternative and better explanationfor the SM fermion mass and mixing hierarchy than the onesconsidered in Refs. [69,78]. To this end we propose a multi-scalar singlet extension of the SU (3)C × SU (3)L × U (1)Xmodel with right-handed neutrinos, where β = − 1√

3and an

extra S3 ⊗ Z3 ⊗ Z ′3 ⊗ Z8 ⊗ Z16 discrete group, that extends

the symmetry of the model and 15 very heavy SU (3)L sin-glet scalar fields are added with the aim to generate viabletextures for the fermion sector, which successfully describethe observed SM fermion mass and mixing pattern. Let usnote that whereas the scalar sector of our model only hasthree SU (3)L scalar triplets and 15 SU (3)L scalar singlets,the scalar sector of the S3 flavor 3-3-1 model of Ref. [69] hassix SU (3)L scalar triplets and four SU (3)L scalar antisextets.Whereas in the model of Ref. [69], the quark mixing matrixis equal to the identity, in our model the quark mixing matrixis in excellent agreement with the low energy quark flavordata. In our model, the obtained physical observables in thequark and lepton sector are consistent with the experimentaldata. Our model at low energies reduces to the 3-3-1 modelwith right-handed neutrinos, where β = − 1√

3. Furthermore,

our current model does not include the U (1)L new leptonglobal symmetry presented in the S3 flavor 3-3-1 model ofRef. [69]. Unlike the S3 flavor 3-3-1 model of Ref. [69],in our current 3-3-1 model, the charged fermion mass andquark mixing pattern can successfully be accounted for, byhaving all Yukawa couplings of order unity and arises fromthe breaking of the Z3⊗Z ′

3⊗Z8⊗Z16 discrete group at veryhigh energy, triggered by SU (3)L scalar singlets acquiringvacuum expectation values much larger than the TeV scale.Despite our current model has more SU (3)L scalar singletsthan the model that two of us have recently proposed in Ref.[78], our current model addresses both the quark and leptonsectors and does not have the aforementioned drawbacks ofthe model of Ref. [78]. Because of the aforementioned rea-sons, our current model represents an important improvementover the previously studied scenarios [69,78]. The particu-lar role of each additional scalar field and the correspond-ing particle assignments under the symmetry group of themodel under consideration are explained in detail in Sect.2. The model we are building with the aforementioned dis-crete symmetries, preserves the content of particles of the3-3-1 model with β = − 1√

3, but we add 15 additional very

heavy SU (3)L singlet scalar fields, with quantum numbersthat allow to build Yukawa terms invariant under the local anddiscrete groups. This generates the right textures that suc-cessfully account for SM fermion masses and mixings. Weassume that the Majorana neutrinos have very small masses,implying that the small active neutrino masses are generatedvia an inverse seesaw mechanism. This mechanism for thegeneration of the light active neutrino masses differs fromthe one implemented in the S3 flavor 3-3-1 model of Ref.

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[69], where the light active neutrinos get their masses fromtype I and type II seesaw mechanisms.

The paper is outlined as follows. In Sect. 2 we explainsome theoretical aspects of the 3-3-1 model with β = − 1√

3and its particle content, as well as the particle assignmentsunder doublet and singlet S3 representations, in particularin the fermionic and scalar sector. The low energy scalarpotential of our model is discussed in Sect. 2.2. In Sect. 3 wefocus on the discussion of lepton masses and mixing and giveour corresponding results. In Sect. 4, we present our resultsin terms of quark masses and mixing, which is followed bya numerical analysis. Conclusions are given Sect. 5. In theappendices we present several technical details: Appendix Agives a brief description of the S3 group; Appendix B shows adiscussion of the stability conditions of the low energy scalarpotential.

2 The model

2.1 Particle content

The first 3-3-1 model with right-handed Majorana neutrinosin the SU (3)L lepton triplet was considered in [160]. How-ever, that model cannot describe the observed pattern of SMfermion masses and mixings, due to the unexplained hier-archy among the large number of Yukawa couplings in themodel. Below we consider a SU (3)C ⊗ SU (3)L ⊗U (1)X ⊗S3 ⊗ Z3 ⊗ Z ′

3 ⊗ Z8 ⊗ Z16 multiscalar singlet extension of the3-3-1 model with right-handed neutrinos, which successfullydescribes the SM fermion mass and mixing pattern. In ourmodel the full symmetry G is spontaneously broken in threesteps as follows:

G = SU (3)C ⊗ SU (3)L ⊗U (1)X ⊗ S3 ⊗ Z3 ⊗ Z ′3 ⊗ Z8 ⊗ Z16

int−−−→SU (3)C ⊗ SU (3)L ⊗U (1)X ⊗ Z3vχ−→SU (3)C ⊗ SU (2)L ⊗U (1)Y

vη,vρ−−−→SU (3)C ⊗U (1)Q , (1)

where the hierarchy vη, vρ � vχ � int among the sym-metry breaking scales is fulfilled.

The electric charge in our 3-3-1 model is defined as [92]

Q = T3 − 1√3T8 + X I, (2)

where T3 and T8 are the SU (3)L diagonal generators, I is the3 × 3 identity matrix and X the U (1)X charge.

Two families of quarks are grouped in a 3∗ irreducible rep-resentations (irreps), as required from the SU (3)L anomalycancellation. Furthermore, from the quark colors, it followsthat the number of 3∗ irreducible representations is six. The

other family of quarks is grouped in a 3 irreducible represen-tation. Moreover, there are six 3 irreps taking into accountthe three families of leptons. Consequently, the SU (3)L rep-resentations are vector like and do not contain anomalies.The quantum numbers for the fermion families are assignedin such a way that the combination of the U (1)X represen-tations with other gauge sectors is anomaly free. Therefore,the anomaly cancellation requirement implies that quarks areunified in the following (SU (3)C , SU (3)L ,U (1)X ) left- andright-handed representations:

Q1,2L =

⎛

⎝D1,2

−U1,2

J1,2

⎞

⎠

L

: (3, 3∗, 0), Q3L =

⎛

⎝U3

D3

T

⎞

⎠

L

: (3, 3, 1/3),

(3)

D1,2,3R : (3, 1, −1/3),

J1,2R : (3, 1, −1/3),

U1,2,3R : (3, 1, 2/3),

TR : (3, 1, 2/3).(4)

Here UiL and Di

L (i = 1, 2, 3) are the left-handed up- anddown-type quarks in the flavor basis. The right-handed SMquarks Ui

R and DiR (i = 1, 2, 3) and right-handed exotic

quarks TR and J 1,2R are assigned into SU (3)L singlets repre-

sentations, so that theirU (1)X quantum numbers correspondto their electric charges.

Furthermore, cancellation of anomalies implies that lep-tons are grouped in the following (SU (3)C , SU (3)L ,U (1)X )

left- and right-handed representations:

L1,2,3L =

⎛

⎝ν1,2,3

e1,2,3

(ν1,2,3)c

⎞

⎠

L

: (1, 3,−1/3), (5)

eR : (1, 1,−1),

N 1R : (1, 1, 0),

μR : (1, 1,−1),

N 2R : (1, 1, 0),

τR : (1, 1,−1),

N 3R : (1, 1, 0).

(6)

where νiL and eiL (eL , μL , τL ) are the neutral and chargedlepton families, respectively. Let us note that we assign theright-handed leptons as SU (3)L singlets, which implies thattheir U (1)X quantum numbers correspond to their electriccharges. The exotic leptons of the model are three neutralMajorana leptons (ν1,2,3)cL and three right-handed Majorana

leptons N 1,2,3R (A recent discussion of double and inverse

seesaw neutrino mass generation mechanisms in the contextof 3-3-1 models can be found in Ref. [182]).

The scalar sector the 3-3-1 models includes: three 3’sirreps of SU (3)L , where one triplet χ gets a TeV scalevaccuum expectation value (VEV) vχ , which breaks theSU (3)L ⊗ U (1)X symmetry down to SU (2)L ⊗ U (1)Y ,thus generating the masses of non-SM fermions and non-SM gauge bosons; and two light triplets η and ρ acquiringelectroweak scale VEVs vη and vρ , respectively, and thusproviding masses for the fermions and gauge bosons of theSM.

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Regarding the scalar sector of the minimal 331 model, weassign the scalar fields in the following [SU (3)L ,U (1)X ]representations:

χ =⎛

⎜⎝

χ01

χ−2

1√2(υχ + ξχ ± iζχ )

⎞

⎟⎠ : (3,−1/3),

ρ =⎛

⎜⎝

ρ+1

1√2(υρ + ξρ ± iζρ)

ρ+3

⎞

⎟⎠ : (3, 2/3),

η =⎛

⎜⎝

1√2(υη + ξη ± iζη)

η−2

η03

⎞

⎟⎠ : (3,−1/3). (7)

We extend the scalar sector of the minimal 331 model byadding the following 15 very heavy SU (3)L scalar singlets:

σ ∼ (1, 0), φ : (1, 0), ζ : (1, 0),

ϕ j : (1, 0), ξ j : (1, 0),

τ j : (1, 0), � j : (1, 0), j = 1, 2,

�k : (1, 0), k = 1, 2, 3, 4.

(8)

We assign the scalars into S3 doublet, and S3 singlet repre-sentations. The S3 ⊗ Z3 ⊗ Z ′

3 ⊗ Z8 ⊗ Z16 assignments ofthe scalar fields are

η ∼(1, e

2π i3 , 1, 1, 1

), ρ ∼

(1,e− 2π i

3 , 1, 1, 1)

,

χ ∼ (1,1, 1, 1, 1) , σ ∼(1′,1, 1, 1, e− π i

8

)

φ ∼ (1′,1, 1, −i, 1

), ζ ∼ (

1′,1, 1, 1, 1),

ξ ∼ (2,1, 1, −1, 1) , τ ∼(2,1, 1, i

12 , 1

),

ϕ1 ∼(1,e− 2π i

3 , 1, −i, 1)

, ϕ2 ∼(1′,e− 2π i

3 , 1, −i, 1)

,

� ∼(2,e− 2π i

3 , 1, −i, 1)

, �1 ∼(1, 1, e

2π i3 , −1, e

3iπ8

),

�2 ∼(1, 1, e

2π i3 , −1, e

2iπ8

), �3 ∼

(1′, 1, e

2π i3 , −1, e− iπ

7

),

�4 ∼(1, 1, e− 2π i

3 , −1, 1)

.

It has been shown in Ref. [81] that the minimization equa-tions for the scalar potential involving the S3 scalar doublet,imply that the S3 scalar doublets ξ , τ and � can acquire thefollowing VEV pattern:

〈ξ 〉 = vξ (1, 0) , 〈τ 〉 = vτ (1, 1) , 〈�〉 = v� (1, 0) . (9)

The vacuum configuration of a S3 scalar doublet, pointingeither in the (1, 0) or in the (1, 1) S3 directions, has beenconsidered in several S3 flavor models (see for instanceRefs. [78,81,211]). In our model we assume the hierarchyv� << vτ << vξ , between the VEVs of the S3 scalar dou-blets in order to neglect the mixings between these fields

and to treat their scalar potentials independently. Let us notethat mixing angles between those fields are suppressed by theratios of their VEVs, as follows from the method of recursiveexpansion of Ref. [212].

In the lepton sector, we have the following S3 ⊗Z3 ⊗Z ′3 ⊗

Z8 ⊗ Z16 assignments:

L1L ∼

(1,e

2π i3 , 1, i

12 , 1

),

LL =(L2L , L3

L

)∼

(2,e

2π i3 , 1, i

12 , 1

)

eR ∼(1′,e− 2π i

3 , 1, i12 , −1

), μR ∼

(1′,e− 2π i

3 , 1, 1, eπ i4

),

τR ∼(1′,e− 2π i

3 , 1, 1, 1)N1R ∼

(1,e

2π i3 , 1, i

12 , 1

)

NR =(N2R, N3

R

)∼

(2,e

2π i3 , 1, i

12 , 1

), (10)

while the S3 ⊗ Z3 ⊗ Z ′3 ⊗ Z8 ⊗ Z16 assignments for the quark

sector are

QL = (Q1L , Q2L) ∼(2, 1, 1,−1, e− iπ

8

),

Q3L ∼ (1,1, 1, 1, 1) ,

U 1R ∼

(1,e− 2π i

3 , e2π i

3 , 1, e6iπ

8

),

U 2R ∼

(1′,e− 2π i

3 , e2π i

3 , 1, e2iπ

8

),

U 2R ∼

(1,e− 2π i

3 , 1, 1, 1)

,

D1R ∼

(1, e− 2π i

3 , 1, 1, e5iπ

8

),

D2R ∼

(1, e− 2π i

3 , e− 2π i3 ,−1, e

3iπ8

),

D3R ∼

(1′, e− 2π i

3 , 1,−1, 1)

,

TR ∼ (1′,1, 1, 1, 1

), J 1

R ∼ (1′,1, 1, 1,−1

),

J 2R ∼ (

1′,1, 1, 1,−i). (11)

In the following we explain the role each discrete group fac-tors of our model. The S3, Z3, Z ′

3, and Z8 discrete groupsreduce the number of the SU (3)C ⊗ SU (3)L ⊗ U (1)Xmodel parameters. This allow us to get viable textures forthe fermion sector that successfully describe the prevailingpattern of fermion masses and mixings, as we will show inSects. 3 and 4. Let us note that we use the S3 discrete groupsince it is the smallest non-Abelian group that has been con-siderably studied in the literature. It is worth mentioning thatthe SU (3)L scalar triplets are assigned to a S3 trivial sin-glet representation, whereas the SU (3)L scalar singlets areaccommodated into three S3 doublets, three S3 trivial sin-glets and three S3 non-trivial singlets. The Z3 and Z8 sym-metries determines the allowed entries of the charged leptonmass matrix. Furthermore, the Z3 symmetry distinguishes theright-handed exotic quarks, being neutral under Z3 from theright-handed SM quarks, charged under this symmetry. Notethat SM right-handed quarks are the only quark fields trans-

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634 Page 6 of 19 Eur. Phys. J. C (2016) 76 :634

forming non trivially under the Z3 symmetry. This results inthe absence of mixing between SM quarks and exotic quarks.Consequently, the Z3 symmetry is crucial for decoupling theSM quarks from the exotic quarks. Besides that, the Z ′

3 sym-metry selects the allowed entries of the SM quark mass matri-ces. Besides that, the Z8 symmetry separates the S3 scalardoublets participating in the quark Yukawa interactions fromthose ones participating in the charged lepton and neutrinoYukawa interactions. The Z16 symmetry generates the hierar-chy among charged fermion masses and quark mixing anglesthat yields the observed charged fermion mass and quarkmixing pattern. It is worth mentioning that the properties ofthe ZN groups imply that the Z16 symmetry is the smallestcyclic symmetry that allows one to build the Yukawa term

L1LρeR

σ 8

8 of dimension 12 from a σ 8

8 insertion on the L1LρeR

operator, crucial to get the required λ8 suppression (whereλ = 0.225 is one of the Wolfenstein parameters) needed tonaturally explain the smallness of the electron mass.

Now let us briefly comment on a possible large discretesymmetry group that could be used to embed the S3 ⊗ Z3 ⊗Z ′

3 ⊗ Z8 ⊗ Z16 discrete symmetry of our model. Consideringthat the discrete group�

(6N 2

)is isomorphic to (ZN×Z ′

N )�

S3 [39] and the fact the Z24 discrete group is the smallestcyclic group that contains the Z3 and Z8 symmetries and theZ ′

3 symmetry is contained in the Z ′24 group, it follows that

the S3 ⊗ Z3 ⊗ Z ′3 ⊗ Z8 ⊗ Z16 discrete group of our model

can be embedded in the �(6N 2

) = � (3456) discrete group(where N = 24). It would be interesting to implement the�

(6N 2

)discrete symmetry in the 331 model and to study its

implications on fermion masses and mixings. This requirescareful studies that are beyond the scope of the present paperand will be done elsewhere.

With the aforementioned field content of our model, therelevant quark and lepton Yukawa terms invariant under thegroup G, take the form

L(Q)Y = y(U )

33 Q3LηU3

R + y(U )23 Q

2Lρ∗U3

Rξσ

2

+y(U )22 Q

2Lρ∗UR

ξσ 3

4

+ y(U )11 Q

1Lρ∗UR

ξσ 7

8

+ y(D)33 Q

3LρD3

Rσ 2�2

3 + y(D)22 QLη∗D2

Rξ�3σ 3

5

+ y(D)12 QLη∗D2

Rξ�4σ 4

6

+ y(D)13 QLη∗D3

Rξσ 4�1

6 + y(D)11 QLη∗D1

Rξσ 6

7

+ y(T )Q3LχTR + y(J )

1 Q1Lχ∗ J1

R + y(J )2 Q

2Lχ∗ J2

R + H.c.,

(12)

− L(L)Y = h(L)

1ρeL1LρeR

σ 8

8 + h(L)1ρμ

(LLρτ

)1 μR

σ 2

3

+ h(L)2ρμ

(LLρτ

)1′ μR

σ 2ζ

4

+ h(L)1ρτ

(LLρτ

)1 τR

1

+ h(L)

2ρτ

(LLρτ

)1′ τR

ζ

2

+ h(L)1χ

(LLχNR

)1 + h(L)

3χ L1LχN 1

R

+ 1

2h1N

(N RN

CR

)

1ϕ1

+ 1

2h2N N

1RN

1Rϕ1 + 1

2h3N

(N RN

CR

)

1′ ϕ2

+ h(1)ρ εabc

(LaL

(LCL

)b)

1′

(ρ∗)c φ

+ h(2)ρ εabc

(LaL

(L1CL

)b (ρ∗)c �

)

1

1

+ h(3)ρ εabc

((L

1L

)a (LCL

)b (ρ∗)c �

)

1

1

+ H.c.,

(13)

where the dimensionless couplings in Eqs. (12) and (13) areO(1) parameters.

Considering that the charged fermion mass and quark mix-ing pattern arises from the breaking of the Z3⊗Z ′

3⊗Z8⊗Z16

discrete group, we set the VEVs of the SU (3)L singlet scalarsσ , ζ , φ, τ j , � j , ξ j ( j = 1, 2) and �k (k = 1, 2, 3, 4) scalarsinglets, as follows:

vφ ∼ v� ∼ λ5 � vτ

= λ3 � vσ = vζ = vξ = v�k = int = λ ,

k = 1, 2, 3, 4, (14)

where λ = 0.225 is one of the parameters of the Wolfen-stein parametrization and the cutoff of our model. Let usnote that the SU (3)L singlet scalar fields σ , ζ , ξ j ( j = 1, 2)and �k (k = 1, 2, 3, 4) having the VEVs of the same order ofmagnitude are the ones that appear in the SM charged fermionYukawa terms, thus playing an important role in generatingthe SM charged fermion masses and quark mixing angles.Regarding the SU (3)L singlet scalar fields τ j ( j = 1, 2),which participates in the charged lepton Yukawa interactions,we assume that it acquires a VEV, much smaller than λ (weset its VEV as λ3 ) in order to suppress its mixing with the S3

scalar doublet ξ , which allows us to treat their scalar poten-tials independently. Because of the reason mentioned above,we have also assumed that the S3 scalar doublet �, whichappears in the Dirac neutrino Yukawa terms, acquires a VEVmuch smaller than λ3 , which we set close to λ5 . Further-more, in order to have the neutrino sector model parametersof the same order, we have assumed that vφ ∼ v�. As previ-ously mentioned, the aforementioned hierarchy between theVEVs of the S3 scalar doublets ξ , τ and � allows us to treattheir scalar potentials independently, thus providing a more

123

Eur. Phys. J. C (2016) 76 :634 Page 7 of 19 634

natural justification for their chosen VEV patterns given inEq. (9) as natural solutions of the scalar potential minimiza-tion equations for the whole region of parameter space.

As we will explain in the following, we are going to imple-ment an inverse seesaw mechanism for the generation of thelight active neutrino masses. To implement an inverse see-saw mechanism, we need very light right-handed Majorananeutrinos, which implies that the SU (3)L singlet scalars hav-ing Yukawa interactions with those neutrinos should acquirevery small vacuum expectation values, much smaller than thescale of breaking of the SM electroweak symmetry. Becauseof this reason, we further assume that the SU (3)L scalar sin-glets ϕ j ( j = 1, 2) giving masses to the right handed Majo-rana neutrinos have VEVs much smaller than the electroweaksymmetry breaking scale, then providing small masses tothese Majorana neutrinos, and thus giving rise to an inverseseesaw mechanism of active neutrino masses. Therefore, wehave the following hierarchy among the VEVs of the scalarfields in our model:

vϕ1 ∼ vϕ2 � vρ ∼ vη ∼ v � vχ � vφ ∼ v� � vτ � int .

(15)

In summary, for the reasons mentioned above and consid-ering a very high model cutoff � vχ , we set the vac-uum expectation values (VEVs) of the SU (3)L scalar sin-glets at a very high energy, much larger than vχ ≈ O(1)

TeV, with the exception of the VEVs of ϕ j , � j ( j = 1, 2),taken to be much smaller than the electroweak symmetrybreaking scale v = 246 GeV. It is noteworthy that theSU (3)C ⊗SU (3)L⊗U (1)X ⊗Z3⊗Z ′

3⊗Z8⊗Z16 symmetryis broken down to SU (3)C ⊗ SU (3)L ⊗U (1)X ⊗ Z3, at thescale int , by the vacuum expectation values of the SU (3)Lsinglet scalar fields σ , ζ , ξ j and �k (k = 1, 2, 3, 4).

It is worth mentioning that in order that the small VEVsof the SU (3)L scalar singlets ϕ j ( j = 1, 2) be stable underradiative corrections, a Veltmann condition that connects acombination of the quartic couplings of the scalar potentialthat involve a pair of these scalar fields with the remain-ing ones and the combination of the Yukawa couplings ofthese scalar singlets with the right-handed Majorana neutri-nos, has to be fulfilled. That Veltmann condition will ariseby requiring the cancellation of the quadratically divergentscalar and fermionic contributions, contributions that inter-fere destructively. The aforementioned Veltmann conditionwill constrain the quartic scalar couplings of the scalar inter-actions involving a pair of the scalar fields that acquire verysmall VEVs. The resulting constraints on these quartic scalarcouplings will not affect neither the fermions masses andmixings nor the flavor changing top quark decays. Havingthe VEVs of the scalar fields of our model stable under radia-tive corrections in the whole region of parameter space, willrequire one to embed our model in a warped five dimensional

framework or to implement supersymmetry. This requirescareful studies which are left beyond the scope of the presentpaper.

2.2 Low energy scalar potential

The renormalizable low energy scalar potential of the modeltakes the form

VH = μ2χ (χ†χ) + μ2

η(η†η) + μ2

ρ(ρ†ρ)

+ f(ηiχ jρkε

i jk + H.c.)

+ λ1(χ†χ)(χ†χ)

+ λ2(ρ†ρ)(ρ†ρ) + λ3(η

†η)(η†η) + λ4(χ†χ)(ρ†ρ)

+ λ5(χ†χ)(η†η)

+ λ6(ρ†ρ)(η†η) + λ7(χ

†η)(η†χ) + λ8(χ†ρ)(ρ†χ)

+ λ9(ρ†η)(η†ρ). (16)

After the symmetry breaking, it is found that the scalar masseigenstates are connected with the weak scalar states by thefollowing relations:

(G±

1H±

1

)= RβT

(ρ±

1η±

2

),

(G0

1A0

1

)= RβT

(ζρ

ζη

),

(H0

1h0

)= RαT

(ξρ

ξη

),

(17)(G0

2H0

2

)= R

(χ0

1η0

3

),

(G0

3H0

3

)= R

(ζχ

ξχ

),

(G±

2H±

2

)= R

(χ±

2ρ±

3

),

(18)

with

RαT (βT ) =(

cos αT (βT ) sin αT (βT )

− sin αT (βT ) cos αT (βT )

), R =

(−1 00 1

),

(19)

where tan βT = vη/vρ , and tan 2αT = M21 /(M2

2 −M23 ) with:

M21 = 4λ6vηvρ + 2

√2 f vχ ,

M22 = 4λ2v

2ρ − √

2 f vχ tan βT , (20)

M23 = 4λ3v

2η − √

2 f vχ/ tan βT .

The low energy physical scalar spectrum of our modelincludes: four massive charged Higgs (H±

1 , H±2 ), one CP-

odd Higgs (A01), three neutral CP-even Higgs (h0, H0

1 , H03 )

and two neutral Higgs (H02 , H

02) bosons. The scalar h0 is

identified with the SM-like 126 GeV Higgs boson found atthe LHC. It it noteworthy that the neutral Goldstone bosons

G01, G0

3, G02, G

02 are associated to the longitudinal compo-

nents of the Z , Z ′, K 0, and K0gauge bosons, respectively.

Furthermore, the charged Goldstone bosons G±1 and G±

2 areassociated to the longitudinal components of the W± and K±gauge bosons, respectively.

123

634 Page 8 of 19 Eur. Phys. J. C (2016) 76 :634

3 Lepton masses and mixings

From Eqs. (9), (13), and (14) and using the product rules ofthe S3 group given in Appendix A, it follows that the massmatrix for charged leptons is

Ml =⎛

⎜⎝a(l)

11 λ8 0 0

0 a(l)22 λ5 a(l)

23 λ3

0 a(l)32 λ5 a(l)

33 λ3

⎞

⎟⎠

v√2. (21)

Since the charged lepton mass hierarchy arises from thebreaking of the Z3 ⊗ Z8 ⊗ Z16 discrete group and in order tosimplify the analysis, we consider a scenario of approximateuniversality in the dimensionless SM charged lepton Yukawacouplings, as follows:

a(l)32 = a(l)

4 , a(l)23 = a(l)

4 e−iα (22)

where a(l)11 , a(l)

22 , a(l)33 and a(l)

4 are assumed to be real O(1)

parameters.The matrix MlM

†l is diagonalized by a rotation matrix Rl

according to

R†l MlM

†l Rl = diag

(me,mμ,mτ

),

Rl =⎛

⎝1 0 00 cos θl − sin θl e−iα

0 sin θl eiα cos θl

⎞

⎠ ,

tan θl −a(l)4

a(l)33

, cos θl a(l)33√(

a(l)33

)2 +(a(l)

4

)2,

sin θl − a(l)4√(

a(l)33

)2 +(a(l)

4

)2, (23)

where from Eq. (21) it follows that the charged lepton massesare approximately given by

me = a(l)11 λ8 v√

2, mμ

∣∣∣∣a

(l)22a

(l)33 −

(a(l)

4

)2∣∣∣∣

√(a(l)

33

)2 +(a(l)

4

)2λ5 v√

2,

mτ √(

a(l)33

)2 +(a(l)

4

)2λ3 v√

2. (24)

Note that the charged lepton masses are connected withthe electroweak symmetry breaking scale v = 246 GeVby their scalings with powers of the Wolfenstein parameterλ = 0.225, with O(1) coefficients. This is consistent withour previous assumption made in Eq. (14) regarding the sizeof the VEVs for the SU (3)L singlet scalars appearing in thecharged fermion Yukawa terms. Furthermore, it is notewor-thy that the mixing angle θl in the charged lepton sector islarge, which gives rise to an important contribution to the

leptonic mixing matrix, coming from the mixing of chargedleptons.

In the concerning to the neutrino sector, the followingneutrino mass terms arise:

−L(ν)mass = 1

2

(νCL νR NR

)Mν

⎛

⎝νLνCRNCR

⎞

⎠ + H.c, (25)

where the S3 discrete flavor group constrains the neutrinomass matrix to be of the form

Mν =⎛

⎝03×3 MD 03×3MT

D 03×3 Mχ

03×3 MTχ MR

⎞

⎠ , MD = vξ vφvρ√2 2

⎛

⎝0 a 0

−a 0 b0 −b 0

⎞

⎠ ,

Mχ = h(L)1χ

vχ√2

⎛

⎝x 0 00 1 00 0 1

⎞

⎠ , MR = h1N vϕ1

⎛

⎝1 0 00 y z0 z y

⎞

⎠ ,

a = h(2)ρ − h(3)

ρ , b = 2h(1)ρ , x = h(L)

2χ

h(L)1χ

,

y = h2N

h1N, z = h3Nvϕ2

h1Nvϕ1

. (26)

Since the SU (3)L scalar singlets ϕ j ( j = 1, 2) havingYukawa interactions with the right-handed Majorana neutri-nos acquire VEVs much smaller than the electroweak sym-metry breaking scale, these Majorana neutrinos are very light,so that the active neutrinos get small masses via the inverseseesaw mechanism.

As shown in detail in Ref. [182], the full rotation matrixis approximately given by

U =

⎛

⎜⎜⎝

Vν B3Uχ B2UR

− (B†2 +B†

3 )√2

Vν(1−S)√

2Uχ

(1+S)√2UR

− (B†2 −B†

3 )√2

Vν(−1−S)√

2Uχ

(1−S)√2UR

⎞

⎟⎟⎠ , (27)

where

S = − 1

2√

2h(L)χ vχ

MR, B2 B3 1

h(L)χ vχ

M∗D, (28)

and the physical neutrino mass matrices are

M (1)ν = MD

(MT

χ

)−1MRM

−1χ MT

D, (29)

M (2)ν = −MT

χ + 1

2MR, M (3)

ν = MTχ + 1

2MR, (30)

where M (1)ν corresponds to the active neutrino mass matrix

whereas M (2)ν and M (3)

ν are the exotic Dirac neutrino massmatrices. Note that the physical eigenstates include threeactive neutrinos and six exotic neutrinos. The exotic neu-trinos are pseudo-Dirac, with masses ∼ ±MT

χ and a smallsplitting MR . Furthermore, Vν , UR and Uχ are the rotation

123

Eur. Phys. J. C (2016) 76 :634 Page 9 of 19 634

matrices which diagonalize M (1)ν , M (2)

ν and M (3)ν , respec-

tively.Furthermore, as follows from Eq. (27), we can connect the

neutrino fields νL = (ν1L , ν2L , ν3L)T , νCR = (νC1R, νC2R, νC3R

)

and NCR = (

NC1R, NC

2R, NC3R

)with the neutrino mass eigen-

states by the following approximate relations:

⎛

⎝νLνCRNCR

⎞

⎠

⎛

⎜⎜⎝

Vνξ(1)L

1√2Uχξ

(2)L + 1√

2URξ

(3)L

− 1√2Uχξ

(2)L + 1√

2URξ

(3)L

⎞

⎟⎟⎠ ,

ξ( j)L =

⎛

⎜⎝

ξ( j)1L

ξ( j)2L

ξ( j)3L

⎞

⎟⎠ , j = 1, 2, 3, (31)

where ξ(1)kL , ξ

(2)kL and ξ

(3)kL (k = 1, 2, 3) are the three active

neutrinos and six exotic neutrinos, respectively.From Eq. (29) it follows that the light active neutrino mass

matrix is given by

M (1)ν = mν

⎛

⎝a2 κab −abκab c2 −κb2

−ab −κb2 b2

⎞

⎠ ,

mν = h1Nvϕ1v2ξ v

2φv2

ρ(h(L)

1χ

)2v2χ 4

, κ = z

y,

c2 = b2 + a2

x2y. (32)

Let us note that the smallness of the active neutrino massesarises from their scaling with inverse powers of the highenergy cutoff as well as from their linear dependence onthe very small VEVs of the SU (3)L singlets ϕ j ( j = 1, 2),assumed to be of the same order of magnitude.

Considering that the orders of magnitude of the SM par-ticles and new physics yield the constraints vχ � 1 TeV andv2η + v2

ρ = v2 and taking into account our assumption thatthe dimensionless lepton Yukawa couplings areO(1) param-eters, from Eq. (32) and the relations vξ = λ , vφ ∼ λ5 ,vρ ∼ 100 GeV, vχ ∼ 1 TeV, we see that the mass scale forthe light active neutrinos satisfies mν ∼ 10−10vϕ . Conse-quently, taking mν ∼ 50 meV, we find for the VEV vϕ1 ofthe singlet scalar ϕ1 the estimate

vϕ1 ∼ 0.5 GeV. (33)

In the following we proceed to fit the lepton sector modelparameters mν , a(l)

11 , a(l)22 , a(l)

33 , a(l)4 , a, b, c and κ to repro-

duce the experimental values for the physical observables ofthe lepton sector, i.e., the three charged lepton masses, thetwo neutrino mass squared splittings and the three leptonicmixing angles. To this end, we fix mν = 50 meV and wevary the parameters a(l)

11 , a(l)22 , a(l)

33 , a(l)4 , a, b, c and κ to fit

Table 3 Model and experimental values of the charged lepton masses,neutrino mass squared splittings, and leptonic mixing parameters forthe normal (NH) and inverted (IH) mass hierarchies. Model values forJarlskog invariant and CP-violating phase

Observable Model value Experimental value

me(MeV ) 0.487 0.487

mμ(MeV ) 102.8 102.8 ± 0.0003

mτ (GeV ) 1.75 1.75 ± 0.0003

�m221(10−5eV2) (NH) 7.60 7.60+0.19

−0.18

�m231(10−3eV2) (NH) 2.48 2.48+0.05

−0.07

sin2 θ12 (NH) 0.323 0.323 ± 0.016

sin2 θ23 (NH) 0.567 0.567+0.032−0.128

sin2 θ13 (NH) 0.0234 0.0234 ± 0.0020

δ (NH) 89.18◦ Unknown

δ (IH) 86.40◦ Unknown

J (NH) 3.46 × 10−2 Unknown

J (IH) 3.49 × 10−2 Unknown

�m221(10−5eV2) (IH) 7.60 7.60+0.19

−0.18

�m213(10−3eV2) (IH) 2.38 2.38+0.05

−0.06

sin2 θ12 (IH) 0.323 0.323 ± 0.016

sin2 θ23 (IH) 0.573 0.573+0.025−0.043

sin2 θ13 (IH) 0.0240 0.0240 ± 0.0019

the charged lepton masses, the neutrino mass squared split-tings �m2

21, �m231 (note that we define �m2

i j = m2i − m2

j )

and the leptonic mixing angles sin2 θ12, sin2 θ13, and sin2 θ23

to their experimental values for normal (NH) and Inverted(IH) neutrino mass hierarchy. The results shown in Table 3correspond to the following best-fit values:

a(l)11 0.42, a(l)

22 1.88, a(l)33 0.67,

a(l)4 0.58, a 0.28, b 0.39

c −0.97, κ 1.11

θl −41.69◦, α −85.99◦, for NH, (34)

a(l)11 0.42, a(l)

22 2.33, a(l)33 0.57,

a(l)4 −0.67, a 0.98, b 0.15

c −0.99, κ −0.05

θl 49.20◦, α −93.60◦, for IH. (35)

Using the best-fit values given above, we get, for NH andIH, respectively, the following neutrino masses:

m1 = 0, m2 ≈ 8.72 meV, m3 ≈ 49.80 meV, for NH,

(36)

m1 ≈ 49.56 meV, m2 ≈ 48.79 meV, m3 = 0, for IH.

(37)

The obtained and experimental values of the observables inthe lepton sector are shown in Table 3. The experimental

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634 Page 10 of 19 Eur. Phys. J. C (2016) 76 :634

0.00 0.01 0.02 0.03 0.040.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

sin2θ13 sin2θ13

sin2θ23

sin2θ12

0.00 0.01 0.02 0.03 0.040.20

0.25

0.30

0.35

0.40

0.000 0.001 0.002 0.003 0.004 0.005 0.0060.0000

0.00002

0.00004

0.00006

0.00008

0.0001

Δm312 [eV2]

Δm212[eV2 ]

Fig. 1 Correlations between sin2 θ23 and sin2 θ13, sin2 θ12 and sin2 θ13,�m2

21 and �m231 for the case of normal hierarchy. The horizonal and

vertical lines are the minimum and maximum values of the leptonic

mixing parameters and neutrino mass squared splittings inside the 3σ

experimentally allowed range

0.00 0.01 0.02 0.03 0.040.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.00 0.01 0.02 0.03 0.040.2

0.4

0.6

0.8

1.0

0.000 0.001 0.002 0.003 0.004 0.005 0.0060.00000

0.00002

0.00004

0.00006

0.00008

0.00010

0.00012

0.00014

sin2θ23

sin2θ12

Δm212[eV2 ]

sin2θ13 sin2θ13 Δm312 [eV2]

Fig. 2 Correlations between sin2 θ23 and sin2 θ13, sin2 θ12 and sin2 θ13,�m2

21 and �m231 for the case of inverted hierarchy. The horizontal and

vertical lines are the minimum and maximum values of the leptonic

mixing parameters and neutrino mass squared splittings inside the 3σ

experimentally allowed range

values of the charged lepton masses, which are given at theMZ scale, have been taken from Ref. [213] (which are similarto those in [214]), whereas the experimental values of theneutrino mass squared splittings and leptonic mixing anglesfor both normal (NH) and inverted (IH) mass hierarchies, aretaken from Ref. [8]. The obtained charged lepton masses,neutrino mass squared splittings, and lepton mixing anglesare in excellent agreement with the experimental data for bothnormal and inverted neutrino mass hierarchies. We found aleptonic Dirac CP-violating phase close to π

2 and a Jarlskoginvariant close to about 3×10−2 for both normal and invertedneutrino mass hierarchy.

In order to study the sensitivity of the obtained values forthe neutrino mass squared splittings, under small variationsaround the best-fit values (maximum variation of +0.2, min-imum of −0.2), we show in Figs. 1 and 2 the correlationsbetween sin2 θ23 and sin2 θ13, sin2 θ12 and sin2 θ13, �m2

21,and �m2

31 for the case of normal and inverted neutrino masshierarchies, respectively. These figures show that a slightvariation from the best-fit values yields for several pointsof the parameter space an important deviation in the valuesof the neutrino mass squared splittings and leptonic mixing

parameters, thus making it difficult to reproduce their exper-imental values, especially for the case of inverted neutrinomass hierarchy. Thus, the solution corresponding to the best-fit point is fine-tuned in the case of inverted neutrino masshierarchy. Addressing this problem requires one to considera discrete flavor group having a triplet irreducible represen-tation, such as, for example A4, S4 and T ′. This will yieldmore predictive textures for the lepton sector thus solving thefine tuning problem. Addressing this issue requires a care-ful investigation beyond the scope of the present paper andwhich is is left for future studies.

Now we determine the effective Majorana neutrino massparameter, which is proportional to the neutrinoless doublebeta (0νββ) decay amplitude. The effective Majorana neu-trino mass parameter is given by

mββ =∣∣∣∣∣∣

∑

j

U 2ekmνk

∣∣∣∣∣∣, (38)

where U 2ej and mνk are the PMNS mixing matrix elements

and the Majorana neutrino masses, respectively.

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Eur. Phys. J. C (2016) 76 :634 Page 11 of 19 634

We predict the effective Majorana neutrino mass parame-ter for both normal and inverted hierarchies:

mββ ={

4 meV for NH,

48 meV for IH.(39)

Our obtained value mββ ≈ 4 meV for the effective Majo-rana neutrino mass parameter in the case of normal hier-archy, is beyond the reach of the present and forthcoming0νββ decay experiments. Concerning the inverted neutrinomass hierarchy, we get the value mββ ≈ 48 for the Majo-rana neutrino mass parameter, which is within the declaredreach of the next-generation bolometric CUORE experiment[215] or, more realistically, of the next-to-next-generationton-scale 0νββ-decay experiments. The current best upperbound on the effective neutrino mass is mββ ≤ 160 meV,

which corresponds to T 0νββ1/2 (136Xe) ≥ 1.6 × 1025 year at

90% C.L, as indicated by the EXO-200 experiment [216].This bound will be improved within a not too far future.The GERDA “phase-II” experiment [217,218] is expectedto reach T 0νββ

1/2 (76Ge) ≥ 2 × 1026 year, which correspondsto mββ ≤ 100 meV. A bolometric CUORE experiment,using 130Te [215], is currently under construction and hasan estimated sensitivity of about T 0νββ

1/2 (130Te) ∼ 1026 years,which corresponds tomββ ≤ 50 meV. Furthermore, there areproposals for ton-scale next-to-next-generation 0νββ exper-iments with 136Xe [219,220] and 76Ge [217,221] claimingsensitivities over T 0νββ

1/2 ∼ 1027 years, which corresponds tomββ ∼ 12–30 meV. For a recent review, see for example Ref.[222]. Consequently, as follows from Eq. (39), our model pre-dicts T 0νββ

1/2 at the level of sensitivities of the next-generationor next-to-next-generation 0νββ experiments.

4 Quark masses and mixing

From Eq. (12) and taking into account that the VEV patternof the SU (3)L singlet scalar fields is described by Eq. (9),with the nonvanishing VEVs set to be equal to λ (being

the cutoff of our model) as shown in Eq. (14), it follows thatthe SM quark mass matrices have the form

MU =⎛

⎜⎝a(U )

1 λ8 0 0

0 a(U )2 λ4 a(U )

4 λ2

0 0 a(U )3

⎞

⎟⎠

v√2,

MD =⎛

⎜⎝a(D)

1 λ7 a(D)4 λ6 a(D)

5 λ6

0 a(D)2 λ5 0

0 0 a(D)3 λ3

⎞

⎟⎠

v√2, (40)

where λ = 0.225 is one of the Wolfenstein parameters,v = 246 GeV the scale of electroweak symmetry breakingand a(U )

i (i = 1, 2, 3, 4) and a(D)j ( j = 1, 2, 3, 4) are O(1)

parameters. From the SM quark mass textures given above,it follows that the Cabibbo mixing as well as the mixing in

Table 4 Model and experimental values of the quark masses and CKMparameters

Observable Model value Experimental value

mu(MeV ) 1.16 1.45+0.56−0.45

mc(MeV ) 641 635 ± 86

mt (GeV ) 174 172.1 ± 0.6 ± 0.9

md (MeV ) 2.9 2.9+0.5−0.4

ms(MeV ) 59.2 57.7+16.8−15.7

mb(GeV ) 2.85 2.82+0.09−0.04

sin θ12 0.225 0.225

sin θ23 0.0407 0.0412

sin θ13 0.00352 0.00351

δ 68◦ 68◦

the 1–3 planes emerges from the down-type quark sector,whereas the up-type quark sector generates the quark mixingangle in the 2–3 planes. Besides that, the low energy quarkflavor data indicates that the CP-violating phase in the quarksector is associated with the quark mixing angle in the 1–3planes, as follows from the standard parametrization of thequark mixing matrix. Consequently, in order to get quarkmixing angles and a CP-violating phase consistent with theexperimental data, we assume that all dimensionless param-eters given in Eq. (40) are real, except for a(D)

5 , taken to becomplex.

Furthermore, the exotic quark masses read

mT = y(T ) vχ√2, mJ 1 = y(J )

1vχ√

2= y(J )

1

y(T )mT ,

mJ 2 = y(J )2

vχ√2

= y(J )2

y(T )mT . (41)

Since the charged fermion mass and quark mixing patternarises from the breaking of the Z3 ⊗ Z ′

3 ⊗ Z8 ⊗ Z16 discretegroup, and in order to simplify the analysis, we adopt a bench-mark where we set a(D)

4 = a(D)1 as well as a(U )

1 = a(U )3 = 1

and a(D)3 = a(U )

2 , motivated by naturalness arguments andby the relationmc ∼ mb, respectively. Then we proceed to fitthe six parameters a(U )

2 , a(U )4 , a(D)

1 , a(D)2 , a(D)

5 and the phaseτ , to reproduce the 10 physical observables of the quark sec-tor, i.e., the six quark masses, the three mixing angles, andthe CP-violating phase. The obtained values for the quarkmasses, the three quark mixing angles, and the CP-violatingphase δ in Table 4 correspond to the best-fit values:

a(U )2 1.43, a(U )

4 0.80, a(D)1 0.58,

a(D)2 0.57,

∣∣∣a(D)5

∣∣∣ 0.44, τ = 68◦. (42)

The obtained quark masses, quark mixing angles and CP-violating phase are consistent with the experimental data. Letus note that despite the aforementioned simplifying assump-

123

634 Page 12 of 19 Eur. Phys. J. C (2016) 76 :634

tions that allow us to eliminate some of the free parameters,a good fit with the low energy quark flavor data is obtained,showing that our model is indeed capable of a very good fitto the experimental data of the physical observables for thequark sector. The obtained and experimental values for thephysical observables of the quark sector are reported in Table4. We use the experimental values of the quark masses at theMZ scale, from Ref. [213] (which are similar to those in[214]), whereas the experimental values of the CKM param-eters are taken from Ref. [9]. We have numerically checkedthat a slight deviation from the best-fit values, keeps all theobtained SM quark masses, with the exception of the bottomquark mass, inside the 3σ experimentally allowed range. Wechecked that small variations around the best-fit values, keepmost of the resulting values of the bottom quark mass insidethe 3σ experimentally allowed range. The values outside the3σ experimentally allowed range are close to the lower andupper experimental bounds of the bottom quark mass. Con-sequently, our model is very predictive for the quark sector.

On the other hand, from the SM quark textures, it followsthat in order to obtain realistic SM quark masses and mix-ing angles without requiring a strong hierarchy among theYukawa couplings, one should have vρ ∼ vη, which impliesthat tan β ∼ O(1). Furthermore, as the h0bb coupling isproportional to sin α

cos β, in order to get a h0bb coupling close

to the SM expectation, we have α ∼ β ± π2 . In the follow-

ing we briefly comment on the phenomenological implica-tions of our model in the concerning to the flavor changingprocesses involving quarks. As previously mentioned, thedifferent Z3 charge assignments for SM and exotic right-handed quark fields imply the absence of mixing betweenthem. The absence of mixings between the SM and exoticquarks will imply that the exotic fermions will not exhibitflavor changing decays into SM quarks and gauge (or Higgs)bosons. After being pair produced they will decay into theSM quarks and the intermediate states of heavy gauge bosons,which in turn decay into the pairs of the SM fermions; seee.g. [228]. The precise signature of the decays of the exoticquarks depends on details of the spectrum and other param-eters of the model. The present lower bounds from the LHCon the masses of the Z ′ gauge bosons in the 3-3-1 modelsreach around 2.5 TeV [229]. One can translate these boundson the order of magnitude of the scale vχ of breaking of theSU (3)C ⊗ SU (3)L ⊗U (1)X ⊗ Z3 symmetry. These exoticquarks can be produced at the LHC via Drell–Yan processesmediated by charged gauge bosons, where the final states willinclude the exotic T quark with a SM down-type quark aswell as any of the exotic J 1 or J 2 quarks with a SM up-typequark. It would be interesting to perform a detailed studyof the exotic quark production at the LHC, the exotic quarkdecay modes and the flavor changing top quark decays. Thisis beyond the scope of this work and is left for future studies.

5 Conclusions

We have constructed an extension of the 3-3-1 model withβ = − 1√

3, based on the extended SU (3)C ⊗ SU (3)L ⊗

U (1)X ⊗ S3 ⊗ Z3 ⊗ Z ′3 ⊗ Z8 ⊗ Z16 symmetry. Our S3 fla-

vor 3-3-1 model, which at low energies reduces to the 3-3-1 model with right-handed neutrinos, where β = − 1√

3, is

in agreement with the current data on SM fermion massesand mixing. The S3, Z3, Z ′

3 and Z8 discrete groups reducethe number of the model parameters. Specifically, the Z3

and Z8 symmetries determine the allowed entries of thecharged lepton mass matrix. Furthermore, the Z3 symme-try decouples the SM quarks from the exotic quarks. TheZ ′

3 symmetry selects the allowed entries of the SM quarkmass matrices. The Z16 symmetry generates the hierarchyamong charged fermion masses and quark mixing anglesthat yields the observed charged fermion mass and quarkmixing pattern. We assumed that the SU (3)L scalar singletshaving Yukawa interactions with the right-handed Majorananeutrinos acquire VEVs much smaller than the electroweaksymmetry breaking scale, then providing very small massesto these Majorana neutrinos, and thus giving rise to an inverseseesaw mechanism of active neutrino masses. The smallnessof the active neutrino masses is attributed to their scalingwith inverse powers of the high energy cutoff as well aswell as by their linear dependence on the very small VEVsof the SU (3)L singlets ϕ j ( j = 1, 2), assumed to be ofthe same order of magnitude. We found for these VEVs theestimate vϕ ∼ 0.5 GeV. The observed hierarchy of the SMcharged fermion masses and quark mixing matrix elementsarises from the breaking of the Z3 ⊗ Z ′

3 ⊗ Z8 ⊗ Z16 discretegroup at very high energy. Furthermore, the model features aleptonic Dirac CP-violating phase close to π

2 and a Jarlskoginvariant close to about 3×10−2 for both normal and invertedneutrino mass hierarchy. In addition, under the assumptionthat the exotic T , J 1 and J 2 quarks are lighter than the H0

2

and H02 neutral Higgs bosons, our model predicts the absence

of the flavor changing neutral exotic quark decays, whichimplies that they can be searched at the LHC via their decayinto the SM quarks and the intermediate states of heavy gaugebosons, which in turn decay into the pairs of the SM fermions;see e.g. [228]. Possible directions for future work along theselines would be to study the constraints on the heavy chargedgauge boson masses in our model arising from the upperbound on the branching fraction for the flavor changing topquark decays, the oblique parameters, the Zbb vertex and theHiggs diphoton signal strength. The heavy exotic quark pro-duction at the LHC may be useful to study. Finally we brieflycomment on a possible large discrete symmetry group thatcould be used to embed the S3 ⊗ Z3 ⊗ Z ′

3 ⊗ Z8 ⊗ Z16 dis-crete symmetry of our model. Considering that the discretegroup �

(6N 2

)is isomorphic to (ZN × Z ′

N ) � S3 [39] and

123

Eur. Phys. J. C (2016) 76 :634 Page 13 of 19 634

the fact the Z24 discrete group is the smallest cyclic groupthat contains both the Z3 and Z8 symmetries, it follows thatthe S3 ⊗ Z3 ⊗ Z ′

3 ⊗ Z8 ⊗ Z16 discrete group of our modelcan be embedded in the �

(6N 2

) = � (3456) discrete group(where N = 24). It would be interesting to implement the�

(6N 2

)discrete symmetry in the 331 model and to study

its implications on fermion masses and mixings. All thesestudies require carefull investigations that we left outside thescope of this work.

Acknowledgements A.E.C.H was supported by Fondecyt (Chile),Grant No. 11130115, by DGIP internal Grant No. 111458 and ProyectoBasal FB0821. R. M. and F.O. were supported by El PatrimonioAutónomo Fondo Nacional de Financiamiento para la Ciencia, la Tec-nolog ía y la Innovación Fransisco José de Caldas programme of COL-CIENCIAS in Colombia. The visit of R.M to Universidad TécnicaFederico Santa María was supported by Fondecyt (Chile), Grant No.11130115.

Open Access This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.Funded by SCOAP3.

Appendix A: The product rules for S3.

The S3 group has three irreducible representations: 1, 1′ and2. Denoting the basis vectors for two S3 doublets as (x1, x2)

T

and (y1, y2)T and y′ a non-trivial S3 singlet, the S3 multipli-

cation rules are [39](x1x2

)

2⊗

(y1y2

)

2= (x1y1 + x2y2)1 + (x1y2 − x2y1)1′

+(x2y2 − x1y1x1y2 + x2y1

)

2, (A1)

(x1x2

)

2⊗ (

y′)1′ =

(−x2y′

x1y′

)

2,

(x ′)

1′ ⊗ (y′)

1′ = (x ′y′)

1 .

(A2)

With these multiplication rules we have to assign to thescalar fields in the S3 irreps and build the corresponding scalarpotential invariant under the symmetry group.

Appendix B: Stability conditions of the low energyscalar potential.

In the following we are going to determine the conditionsrequired to have a stable scalar potential by following themethod described in Ref. [230]. The gauge invariant andrenormalizable low energy scalar potential as a function ofthe fields φ1 = χ , φ2 = ρ and φ3 = η is a linear hermitiancombination of the following terms:

φiφ j , φiφ jφkφl (B1)

where i, j, k, l = φ1, φ2 and φ3. To discuss the stability ofthe potential, its minimum, and its gauge invariance one canmake the following arrangement of the scalar fields by using2 × 2 hermitian matrices:

K(φiφ j ) =(

φ†i φi φ

†i φ j

φ†jφi φ

†jφ j

)

,

= 1

2

(K0(φiφ j)12×2 + Ka(φiφ j)σ

a)

(B2)

where (φiφ j ) = ρη, ρχ, ηχ , σ a (a = 1, 2, 3) are the Paulimatrices and 12×2 is the identity matrix. From the previousexpressions one can build the following bilinear terms asfunctions of the scalar fields:

K0(φiφ j) = φ†i φi + φ

†jφ j ,

Ka(φiφ j) =∑

i, j

(φ

†i φ j

)σ ai j . (B3)

The properties of the potential can be analyzed in terms ofK0(φiφ j) andK(φiφ j) with φiφ j = ρη, ρχ, ηχ in the domain

K0 ≥ 0 y K 20 ≥ K2. Defining κ = K/K0 the potential can

be written as

V = V2 + V4,

V2 =∑

(φiφ j )

K0(φiφ j )J2(φiφ j )(κ),

J2(φiφ j )(κ) = ξ0(φiφ j ) + ξ T(φiφ j )κ (φiφ j ),

V4 =∑

(φiφ j )

K 20(φiφ j )

J4(φiφ j )(κ), (B4)

J4(φiφ j )(κ) = η00(φiφ j ) + 2ηT(φiφ j )

κ (φiφ j )

+ κT(φiφ j )

E(φiφ j )κ (φiφ j ),

where E(φiφ j ) is a 3 × 3 matrix and the functions J2(φiφ j )(κ)

and J4(φiφ j )(κ) are defined in the domain |κ | ≤ 1. The sta-bility of the scalar potential requires that it has to be boundedfrom below. The stability is determined from the behavior ofV in the limit K0 → ∞, i.e.,

J4(φiφ j )(κ) ≥ 0, (B5)

for all |κ | ≤ 1. To impose J4(φiφ j )(κ) to be positively definedit is enough to consider the values of all stationary points inthe domain |κ| < 1 and |κ| = 1. This results in a boundfor η00(φiφ j ), η0(φiφ j ) and E(φiφ j ), which parametrize thequartic terms of the potential included in V4.

For |κ | < 1 the stationary points should satisfy

Eκ (φiφ j ) = −η(φiφ j ), |κ | < 1. (B6)

123

634 Page 14 of 19 Eur. Phys. J. C (2016) 76 :634

For the case where det E �= 0, the following relation isobtained:

J4(φiφ j )(κ)|est = η00(φiφ j ) − ηT(φiφ j )

E−1η(φiφ j ). (B7)

For |κ | = 1 the stationary points are obtained from the func-tion:

F4(φiφ j )(κ) = J4(φiφ j )(κ) + u(1 − κ2), (B8)

where u is a Lagrange multiplier that satisfies the followingcondition:

(E(φiφ j ) − u)κ = −η(φiφ j ),

J4(φiφ j )(κ)|est = u + η00(φiφ j ) (B9)

−ηT(φiφ j )

(E(φiφ j ) − u)−1η(φiφ j ).

The stationary points of J4(φiφ j )(κ) for |κ| ≤ 1 can beobtained from

f(φiφ j )(u) = J4(φiφ j )(κ)|est > 0,

f ′(φiφ j )

(u) > 0. (B10)

Considering that the quartic terms of the scalar potential aredominant when the vacuum expectation values of the scalarfields take large values, these terms will be the most relevantto analyze the stability of the scalar potential. Following themethod described in Ref. [230], we proceed to rewrite thequartic terms of the scalar potential in terms of bilinear com-binations of the scalar fields. To this end, the bilinear com-binations of the scalar fields are included in the followingmatrices:

Kρη =(

ρ†ρ η†ρ

ρ†η η†η

)= 1

2

(K0(ρη)12×2 + Ka(ρη)σ

a) ,

Kρχ =(

ρ†ρ χ†ρ

ρ†χ χ†χ

)= 1

2

(K0(ρχ)12×2 + Ka(ρχ)σ

a) ,

Kηχ =(

η†η χ†η

η†χ χ†χ

)= 1

2

(K0(ηχ)12×2 + Ka(ηχ)σ

a) ,

(B11)

where σ a (a = 1, 2, 3) are the Pauli matrices and 12×2 is the2×2 identity matrix. From the previous expressions, we findthat the bilinear combinations of the scalar fields appearingin Eq. (B11) are given by

K0(ρη) = ρ†ρ + η†η, K0(ρχ) = ρ†ρ + χ†χ,

K0(ηχ) = η†η + χ†χ, (B12)

Ka(ρη) =(ρ†ρ

)σ a

11 +(η†η

)σ a

22 +(ρ†η

)σ a

12 +(η†ρ

)σ a

21,

Ka(ρχ) =(ρ†ρ

)σ a

11 +(χ†χ

)σ a

22 +(ρ†χ

)σ a

12 +(χ†ρ

)σ a

21,

Ka(ηχ) =(η†η

)σ a

11 +(χ†χ

)σ a

22 +(η†χ

)σ a

12 +(χ†η

)σ a

21.

Since the stability of the scalar potential is determined fromits quartic terms, the stationary solutions consistent with astable scalar potential are described by the following func-tions:

fρη (u) = u + E00(ρη) − Ea(ρη)

(E(ρη) − u13×3

)−1ab Eb(ρη),

fρχ (u) = u + E00(ρχ) − Ea(ρχ)

(E(ρχ) − u13×3

)−1ab Eb(ρχ),

fηχ (u) = u + E00(ηχ) − Ea(ηχ)

(E(ηχ) − u13×3

)−1ab Eb(ηχ),

(B13)

where, for the ρ and η fields, we have

E00(ρη) = λ2 + λ3 + λ6

4,

Ea(ρη) = λ2 − λ3

4δa3,

E(ρη) = 1

4

⎛

⎝λ9 0 00 λ9 00 0 λ2 + λ3 − λ6

⎞

⎠ . (B14)

In the same manner, for the multiplets ρ and χ , the expres-sions are

E00(ρχ) = λ1 + λ2 + λ4

4,

Ea(ρχ) = λ1 − λ2

4δa3,

E(ρχ) = 1

4

⎛

⎝λ8 0 00 λ8 00 0 λ1 + λ2 − λ4

⎞

⎠ . (B15)

Similarly, for the η and χ fields, we find

E00(ηχ) = λ1 + λ3 + λ5

4, Ea(ηχ) = λ1 − λ3

4δa3,

E(ηχ) = 1

4

⎛

⎝λ7 0 00 λ7 00 0 λ1 + λ3 − λ5

⎞

⎠ . (B16)

Following Ref. [230], we determine the stability of the scalarpotential from the conditions:

fρη (u) > 0, fρχ (u) > 0, fηχ (u) > 0. (B17)

We use the theorem of stability of the scalar potential ofRef. [230] to determine the stability conditions of the scalarpotential. To this end, the condition fρη (u) > 0 is analyzedfor the set of values of u which include the 0, (since f ρη (0) >

0) the roots u(1)ρη and u(2)

ρη of the equation f ρη (u) = 0 and

the eigenvalues E (a)(ρη) of the matrix E(ρη) where fρη(E

(a)(ρη))

is finite and f ρη(E(a)(ρη)) ≥ 0. We proceed in a similar way

when analyzing the conditions fρχ (u) > 0 and fηχ (u) > 0.

123

Eur. Phys. J. C (2016) 76 :634 Page 15 of 19 634

Therefore, the scalar potential is stable when the followingconditions are fulfilled:

λ1 > 0, λ2 > 0, λ3 > 0,

λ4 + λ8 ≷ 2√

λ1λ2, λ4 + λ8 ≷ λ1 + λ2,

2√

λ1λ2 ≷ λ4, λ1 + λ2 ≷ λ4,

λ5 + λ7 ≷ 2√

λ1λ3, λ5 + λ7 ≷ λ1 + λ3,

2√

λ1λ3 ≷ λ5, λ1 + λ3 ≷ λ5,

λ6 + λ9 ≷ 2√

λ2λ3, λ6 + λ9 ≷ λ2 + λ3,

2√

λ2λ3 ≷ λ6, λ2 + λ3 ≷ λ6. (B18)

Furthermore, having masses m2H±

1, m2

H01

and m2H0

3positively

defined requires the following condition:

f > 0. (B19)

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